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Moments for a ratio of correlated gamma variatesJ.D. Tubbs aa Department of Mathematical Sciences , University of ArkansasPublished online: 27 Jun 2007.

To cite this article: J.D. Tubbs (1986) Moments for a ratio of correlated gamma variates, Communications in Statistics -Theory and Methods, 15:1, 251-259

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COMMUN. STATIST.-THEORY METH., 1 5 ( 1 ) , 251-259 ( 1986 )

MOMENTS FOR A RATIO OF CORRELATED GA'lMA VARIATES

J . D . Tubbs

Department of K a t h e m a t i c a l S c i e n c e s U n i v e r s i t y of Arkansas

Key Words and P h r a s e s : h y p e r g e o m z t ~ ~ e fz,nctions; Kwmer's i d e n t i t y ; bivar

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In this paper the results are restricted to the moments of

positively correlated gamma distributed variates. Flueck, Holland,

and Lee (1975, 1979) presented some exact distributional results for the ratio, R, of correlated gamma under the Cherian-David-Fix

bivariate gamma distributional structure. Recently, Tubbs and

Smith (1984) obtained comparable results for R using a special case of Jensen's bivariate gamma distributional structure. The

corresponding expressions for the moments of R from the two struc- tures were quite different in their appearance.

This paper investigates the comparison of the respective

finite moments whenever they exist. This comparison is performed

both analytically and numerically. In section 2, the results for each of the specified bivariate gamma structures is reviewed.

Section 3 contains the analytical results for the comparison of

the expressions for the respective moments. The numerical results

are presented in section 4.

2. DISTRZBUTTON AXD NPIENTS OF R - - - - - -.- - . - - - - - -

2.1 Cherian-David-Xix-S tructure

The results in this section are found in Lee, Holland, and

Flueck (1979) and llielke and Flueck (1976). Using the notation of ?Iielke and Flueck, the probability density function for R = X / Y

where X and Y are positively correlated gamma variates is given by

(a)m+n(b)m(c)n where F (a,b,c,d:x,y) = C ----------- 1 (d),+,m!n! X Y ,/xl

MOMENTS FOR RATIO OF CORRELATED GAMMA VARIATES 253

X = U+P, Y = V+P. U, V, and P a r e independent gamma random v a r i a -

b l e s with common s c a l e parameter X and r e s p e c t i v e shape parameters

a-5, B - S , and S(O

254 TUBBS

The purpose of t h i s p a p e r i s t o c o n s i d e r b o t h e x p r e s s i o n s

g i v e n i n e q u a t i o n s ( 2 . 2 ) and (2 .5) and d e t e r m i n e unde r which c o n d i t i o n s t h e two e q n a t i o n s p r o v i d e i d e n t i c a l r e s u l t s . Consid-

e r i n g e q u a t i o n (2 .5 ) and t h e d e f i n i t i o n and p r o p e r t i e s of one and two d i m e n s i o n a l hype rgeomet r i c f u n c t i o n s a l o n g w i t h Kummer's

i d e n t i t y , e q u a t i o n ( 2 . 5 ) can b e r e w r i t t e n a s e i t h e r

Tn o r d e r t o perform a compar ison of t h e e x p r e s s i o n f o r t h e

moments g i v e n b y t h e two mr thods , e q u a t i o n ( 2 . 2 ) w i l l be r e w r i t t e n a s

f o r some c o e f f i c i e n t s 5 . Fur the rmore , e q u a t i o n (2.6) can be j w r i t t e n a s

where

Using u ( 5 ) and vrn(C), i t f o l l o w s t h a t t h e two methods a r e iden - m

t i c a l i f a n d o n l y i f a = b . f o r a l l j ' s . Fur the rmore , s i n c e j J

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MOMENTS FOR RATIO OF CORRELATED GAMMA VARIATES 255

i t fo l lows t h a t (2.2) and (2.6) a r e equa l i f and only i f h m ( j ) = k ( j ) f o r a l l j = O , l , . . . ,m < B . The above n o t a t i o n w i l l b e used

m

i n proving t h e main r e s u l t s a s summarized i n t h e fol lowing theorem.

Theorem: Expressions f o r t h e mth unad jus ted popula t ion moments --

f o r R , t h e r a t i o of c o r r e l a t e d gamma v a r i a t e s given by equa t ions

(2.2) and (2.6) a r e i d e n t i c a l whenever e i t h e r i ) X and Y a r e independent and m E (0,R) o r i i ) m = 1 f o r B > 1.

Proof: i) I f X and Y a r e independent then 5 = 0, which i m p l i e s --

t h e equa t ions (3.1) and (3.2) reduce t o km(0) and hm(0) , respec- t i v e l y , f o r any nonnegat ive i n t e g e r m(m< a ) . From equat Ion (3.31, i t fo l lows t h a t h,(O) = ( U ) ~ ( R ) - ~ . By cons ider ing u m ( 5 ) i n equa- t i o n (3.1) and n o t i n g t h a t (a-5).(5),j = 0 i f 5 = 0 and J j = 0 , 1 , 2 , . . .m-1 and t h a t (a-5) . ( s ) , ~ = f o r j = m and

J 5 = 0, hence

I f m = 1 and 5 > 0 then equa t ions (2.2) and (2.6) a r e iden- t i c a l i f and only if h l ( j ) = k l ( j ) f o r j = 0 , l . It fo l lows from (3.3) t h a t

1 h l ( 0 ) = ~ I R - 1 , h l ( l ) = B(D-~).

From (3.5) it fol lows t h a t

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2 5 6

Since ( x ) ~ = 1 and ( x ) ~ = x, i t i s e a s i l y shown t h a t

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i i ) Suppose t h a t 1 < m < B and t h a t 5 > 0. The two expres- s i o n s a r e i d e n t i c a l whenever k m ( j ) = hm(j ) f o r j = 1 , 2 , . . . , m . Consider t h e c a s e where j = m. Again from (3 .3) i t fol lows t h a t

I n cons ider ing km(m) from equa t ion (3 .5 ) , i t is s u f f i c i e n t t o con- m

s i d e r t h e c o e f f i c i e n t 5 i n t h e expansion (a-5) . (5 ) f o r any J m-j

f i x e d j . Since t h i s c o e f f i c i e n t i s always ( - I ) ' , it fo l lows t h a t

In cons ider ing equa t ions (3.7) and ( 3 . 8 ) , one observes t h a t k (m) i s a f u n c t i o n of t h e parameter a v~hereas hm(m) i s indepen-

m

dent of a . Hence, equa t ions ( 2 . 2 ) and (2.6) a r e n o t i d e n t i c a l f o r a r b i t r a r y parameters a and 6.

The above theorem g i v e s necessary and s u f f i c i e n t condi t ions f o r t h e two methods t o provide i d e n t i c a l r e s u l t s . The next s e c t i o n cons iders a comparison of t h e two methods whenever t h e

methods d i f f e r . The comparison i s made us ing a numerical eval- u a t i o n of t h e two methods f o r a s e l e c t e d s u b s e t of t h e parameter space.

4 . NUMERICAL EVALUATE

In t h i s s e c t i o n , equa t ions (2.2) and (2.6) a r e eva lua ted numer ica l ly u s i n g a s e l e c t e d s u b s e t of t h e parameter space. D

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MOMENTS FOR RATIO OF CORRELATED GAMMA VARIATES

METHOD 1 METHOD 2

4 0 . 2 5 0 . 16 l-I . 11 4 0 . 0 5

1 . @ 0 (1.67 0 . 3 6 0 . 2 0

8 . 7 5 5 . 4 1 3 . 1 9 1.79

I . 3 l 0.96 0 . 6 3 0 . 4 9

0 . 4 2 0.33 0 . 2 6 0 .20

t o . 10 0 . OH 0. 07 0.116

2 . t i2 2 . (:I 8 1.64 1.28

o .60 0 .52 0 .44 O. 3 8

I .87 1.60 1.37 1 . 1 7

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258 TUBBS

Tab le 1 summarizes t h e f i r s t f o u r u n a d j u s t e d p o p u l a t i o n moments. Method 1 r e f e r s t o e q u a t i o n ( 2 . ~ ) and Method 2 r e f e r s t o equa- t i o n ( 2 . 6 ) .

From Tab le 1, one o b s e r v e s t h a t t h e h i g h e r unad jus ted moments a r e d i f f e r e n t f o r t h e two methods whenever t h e v a r i a b l e s X and Y a r e c o r r e l a t e d . However, t h e r e is v e r y l i t t l e d i f f e r e n c e i n t h e f i r s t two moments. From t h e t a b l e one a l s o o b s e r v e s t h a t t h e u n a d j u s t e d moments a r e s i g n i f i c a n t l y a f f e c t e d by t h e p r e s e n c e of c o r r e l a t i o n . Th i s phenomenon was obse rved by Flueck and Hol land (1976) i n e s t i m a t i n g t h e moments of r a t i o s u s i n g r a i n - f a l l d a t a . Although t h e e x p r e s s i o n s f o r t h e s e moments a r e func- t i o n s o f which b i v a r i a t e gamma was used t h e r e s u l t s are q u i t e com- p a r a b l e u s i n g e i t h e r method.

5. CONCLUSIONS AND SUMNARY

Express ions f o r t h e p o s i t i v e i n t e g r a l moments a r e g i v e n f o r

t h e r a t i o of c o r r e l a t e d gamma d i s t r i b u t e d v a r i a b l e s . Two under- l y i n g b i v a r i a t e gamma s t r u c t u r e s were cons ide red . It was shown

t h a t t h e two s t r u c t u r e s p r o v i d e i d e n t i c a l r e s u l t s f o r a l l f i n i t e moments whenever t h e v a r i a b l e s a r e u n c o r r e l a t e d and a r e o n l y i d e n t i c a l f o r t h e f i r s t moment whenever t h e v a r i a b l e s a r e c o r r e l a t e d . A numer ica l e v a l u a t i o n f o r a s e l e c t e d s u b s e t o f t h e

pa ramete r s p a c e i n d i c a t e d t h a t a l t h o u g h t h e h i g h e r moments do d i f f e r a c c o r d i n g t o t h e method used , t h e d i f f e r e n c e s a r e p robab ly i n s i g n i f i c a n t f o r most a p p l i c a t i o n s .

Both methods r e v e a l t h e s i g n i f i c a n t e f f e c t t h a t c o r r e l a t i o n h a s upon t h e p o p u l a t i o n moments. Hence, b o t h methods r e i n f o r c e t h e need f o r e x a c t d i s t r i b u t i o n a l r e s u l t s f o r t h e r a t i o o f gamma d i s t r i b u t e d v a r i a b l e s a s i n d i c a t e d i n t h e s t u d y of r a i n f a l l o r r a i n s e e d i n g exper imen t s a s g i v e n i n F lueck and Hol land (1976).

6 . BIBLIOGRAPHY

Dwivedi, T . D. and Chaubey, Y . P . , (1981) . Moments of a r a t i o of two p o s i t i v e q u a d r a t i c forms i n normal v a r i a t e s . Commun. S t a t i s t . - S i m u l a . Computa., 10, 503-516.

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MOMENTS FOR RATIO OF CORRELATED GAMMA VARIATES 259

Gradshteyn, I. S. and Ryshik, I. M . , (1965). T a b l e s o f I n t e g r a l s , S e r i e s , and P r o d u c t s . New York: Academic P r e s s .

F lueck , J . A. and Hol l and , B . S . , (1976) . R a t i o e s t i m a t o r s and some i n h e r e n t problems i n t h e i r u t i l i z a t i o n . J. Appl. Meteoy., 15, 535-543. -

Flueck , J . A . , Ho l l and , B . S . , and Lee, R., (1975) . D i s t r i b u t i o n of t h e r a t i o of c o r r e l a t e d sums of gamma v a r i a b l e s . Proc. S o c i a l Sc ience S e c t i o n , ASA, Washington, D . C . , pp. 285-291.

J e n s e n , D, R . , (1970) . The j o i n t d i s t r i b u t i o n of q u a d r a t i c forms and r e l a t e d d i s t r i b u t i o n s . A u s t r a l . J. of S t a t i s t . , 12, 13-22.

Lee, R . , Ho l l and , B . S . , and F lueck , J. A . , (1979) . D i s t r i b u t i o n of a r a t i o of c o r r e l a t e d gamma random v a r i a b l e s . SIAM J. Appl. Math., 2, 304-320.

d i e l k e , P. IJ. and F lueck , J. A , , (1976) . D i s t r i b u t i o n s of r a t i o s f o r some s e l e c t e d h i v a r i a t e probability f u n c t i o n s . 1976 Proc. S o c i a l S c i e n c e S e c t i o n , ASA, Washington, D.C. , pp. 608-613.

Tubbs, J. D. and Smith , 0 . E . , (1984) . A n o t e on t h e r a t i o o f p o s i t i v e l y c o r r e l a t e d gamma v a r i a t e s . Commun. S t a t i s t . - T h e o r . Meth. , 14, 13-23.

Received Januahq, 198 5; Revdined Septernbeh, 19ti5.

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